SIN contest: ballistic missile

The problem and solution involving the ballistic missile question is in the attached picture.

2. Relevant equations

3. The attempt at a solution

Why does the first expression have (24)(6) in the denominator, when the second expression has (24)(6)(2) ?

My thought process:
If the missile could teleport to latitude 45 right after being launched from latitude 30, and have the velocity of latitude 30, then I can see why the first expression (without the 2) is valid.

But instead, the missile needs to gradually work its way to having that large of a relative velocity as it moves north. So it goes from 0 relative velocity to the full relative velocity and so dividing by 2 will get the average relative velocity.

Then i had doubts that the relationship between relative velocity and time was linear. I worked out some math and it seemed like a trigonometric relationship with velocity being proportional to cosine(arclength).

Am i confusing myself?
I would appreciate it if someone could clarify this for me.

Thanks for pointing that out, but I'm still confused. Since the missile must travel to latitude 45 doesn't it spend time not gaining as much ground as it potentially could compared to if it were on latitude 45 during the entire 10 minutes (but at the higher speed)?